Certain locally nilpotent varieties of groups

Abstract

Let c ≥ 0, d ≥ 2 be integers and Nc(d) be the variety of groups in which every d-generator subgroup is nilpotent of class at most c. N.D. Gupta asked for what values of c and d is it true that Nc(d) is locally nilpotent? We prove that if c ≤ 2d + 2d-1 - 3 then the variety Nc(d) is locally nilpotent and we reduce the question of Gupta about the periodic groups in Nc(d) to the prime power exponent groups in this variety.